Tuesday, August 07, 2007

Riddles

Swedish Hat Riddle, Figure 1


Members of the Davis lab at Stanford = intellectual heavyweights who love solving math problems and riddles. A few weeks ago, Joe gave me a riddle involving a train station. It puzzled me, but I didn’t think very hard about it until Joe retold the riddle to a Stanford med student – who knew the answer immediately. Suddenly, my UCSF pride was on the line (I am joking), but I still couldn’t solve the riddle until later that evening in the shower. This pattern continued for some time…a riddle would be broached by one person…it would be solved within a few minutes by members of the lab… then various members of the lab gathered in a corner away from my desk (where I sit cross-eyed and drooling in a puddle of my own brain juices) to discuss and verify the answer.

Loyal readers, I am writing down these riddles so that you can test yourself against the very brightest – Joe, Paul, and Amit! Using a cool trick from Craig, please highlight the solutions (in white text) with your mouse to see the answer.

Riddle #1: Stephanie take the Caltrain every afternoon to go home. From the Palo Alto train station, she can either go north to San Francisco or south to Mountain View. Both north and south trains arrive at the station every 10 minutes. Instead of deciding which train to take everyday, Stephanie decides to place her destination in the hands of fate and resolves to arrive at the Palo Alto station at random times between the hours of 4-6 p.m. and to take the first train that reaches the platform (only one platform). However, after a few weeks, Stephanie realizes that 9/10 times, she takes the San Francisco train from Stanford. How can this be?



Answer #1: The northern train always arrives at the station 1 minute before the southern train...every 10 minutes.





Riddle #2: The Mafia sends you into a completely dark room. On the table, there is a full deck of cards, randomly shuffled. The black cards are all face-up, and the red cards are all face-down. You must separate the card deck into two equal piles (26 cards in each pile) and manipulate the deck so that the same number of cards are facing "up" in each pile. You may flip cards over. How do you solve the puzzle?



Answer #2: Divide the shuffled deck into two equal groups. Let's pretend for a second that the first group has 2 black cards and 24 red cards (2 cards "up," and 24 cards "down"). This means that the OTHER group has 24 black cards ("up") and 2 red cards ("down"). Simply flip over ONE pile of 26 cards to achieve 2 piles with the same number of "up" and "down"-facing cards.





Riddle #3: The Davis lab used to have a lot of Swedish post-docs. Let's pretend that the PI was in Sweden when 4 postdocs -- Simon, Fredrik, Bob, and Johan -- ask the PI for lab positions in the States. The PI has 2 black hats and 2 white hats. He distributes the hats randomly to the four postdocs and takes them to the men's restroom (Joe invented this scenario, not me). The man cannot see which hat they are wearing. He places one postdoct in the bathroom stall, and no one can see in or out of this stall. The other 3 postdocs are lined up vertically facing the wall of the stall, and can see the hats in front of them (see Figure 1). The PI says that whoever can tell which hat they are wearing can come to Stanford. There is a LONG PAUSE before one man speaks up. Which man correctly guesses which hat he is wearing (black or white) and why?


Answer #3: The man in the stall cannot see anyone else, he is isolated and therefore cannot tell for certain which hat he is wearing. So forget about the man in the stall. The man closest to the stall is also blind to any hats, so he must also be excluded. At first glance, the man farthest away from the stall should be able to tell which hat he is wearing because he can see the hats of the two men in front of him. This would be the case if he saw two white hats, for example, and can safely say that he is wearing a black hat. However, there is a long silence. The two men in front of him must be wearing a white and a black hat. Then, the man in the middle (of the 3 outside) can deduce that the outside man cannot tell which hat he is wearing...which means that if the man in front of him is wearing a white hat (for example), then the middle man knows that he must be wearing the opposite...a black hat (for example).

2 comments:

Joe said...

So THIS is what you do in lab when I'm not there? GET BACK TO WORK! And take off that silly white hat (or was it black?)!Man, to think I had to sign up for a Google account to post this comment... Joe

Stephanie said...

Aw, man, BUSTED!